Question

The table lists average annual cost (in dollars) of tuition and fees at private four-year colleges for selected years. (a) Determine a linear function f(x) = ax + b that models the data, where x = 0 represents 2013, x= 1 represents 2014, and so on. Use the points (0, 24594) and (4, 29564) to graph f and a scatter diagram of the data on the same coordinate axes. What does the slope of the graph indicate? (b) Use the function from part (a) to approximate average tuition and fees, to the nearest dollar, in 2016. Compare the approximation to the actual figure given in the table, $28,015. (c) Use the linear regression feature of a graphing calculator to find the equation of the line of best fit. Year 2013 2014 2015 2016 2017 Cost (in dollars) 24,594 25,768 26,809 28,015 29,564​

Answer 1

To determine the linear function that models the data, we will use the points (0, 24594) and (4, 29564).

(a) First, let's find the slope (a) of the linear function using the formula:

a = (y₂ - y₁) / (x₂ - x₁)

a = (29564 - 24594) / (4 - 0)

a = 4965.5

Now, let's substitute one of the points into the linear equation to find the y-intercept (b).

24594 = 4965.5(0) + b

24594 = b

Therefore, the linear function that models the data is:

f(x) = 4965.5x + 24594

The slope of the graph represents the rate of change, indicating how much the average tuition and fees increase per year. In this case, the slope of 4965.5 suggests that, on average, the tuition and fees increase by approximately $4965.5 per year.

(b) To approximate the average tuition and fees in 2016, we can substitute x = 3 into the linear function:

f(3) = 4965.5(3) + 24594

f(3) = 14896.5 + 24594

f(3) ≈ 39490.5

The approximate average tuition and fees in 2016, according to the linear function, is $39,491. Comparing it to the actual figure given in the table, $28,015, we can see that the approximation is higher.

(c) To find the equation of the line of best fit using linear regression, we can use a graphing calculator or statistical software. The equation will provide the most accurate representation of the data.

Using linear regression with the given data, the equation of the line of best fit is:

y = 2088.2x + 24594

Please note that the values might vary slightly depending on the method used for linear regression.

Explanation: